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would give him all the invariant expressions of the kind
described* To avoid lengthy explanations the problem has been
stated in terms of equations; actually it was attacked otherwise,
but this is of no importance here.
As this question of invariance is fundamental in modern
scientific thought we shall give three further illustrations of
what it means, none of which involves any symbols or algebra.
Imagine any figure consisting of intersecting straight lines and
curves drawn on a sheet of paper. Crumple the paper in any
way you please without tearing it, and try to think what is the
most obvious property of the figure that is the same before and
after crumpling. Do the same for any figure drawn on a sheet
of rubber, stretching but not tearing the rubber in any compli-
cated manner dictated by whim. In this case it is obvious that
sizes of areas and angles, and lengths of lines, have not remained
'invariant*. By suitably stretching the rubber the straight lines
may be distorted into curves of almost any tortuosity you like,
and at the same time the original curves - or at least some of
them - may be transformed into straight lines. Yet sometHing
about the whole figure has remained unchanged; its very sim-
plicity and obviousness might well cause it to be overlooked,
Ttiis is the order of the points on any one of the lines of the
figure which mark the places where other lines intersect the
given one. Thus, if moving the pencil along a given line from
A to C, we had to pass over the point B on the line before the
figure was distorted, we shall have to pass over B in going from
A to C after distortion, The order (as described) is an invariant
under the particular transformations which crumpled the sheet
of paper into a crinkly ball, say, or which stretched the sheet
of rubber.*
This illustration may seem trivial, but anyone who has read
a non-mathematical description of the intersections of 'world-
lines* irbgeneral relativity, and who recalls that an intersection
of two such lines marks a physical 'point-eoertf, will see that
what we have been discussing is of the same stuff as one of our
pictures of the physical universe. The mathematical machinery
powerful enough to handle such complicated 'transformations*
and actually to produce the invariants was the creation of